On the limit-classifications of even and odd-order formally symmetric differential expressions

Original · EN

In this paper we consider the formally symmetric differential expression M[·] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient space D(T)/D(T) associated with M[·] in terms of the behaviour of the determinants equation* r,s∈ Nₙ [[fᵣgₛ](∞)] equation* where 1≤ n≤ (order of the expression + 1); here [fg](∞) = ₓ→∞[fg](x), where [fg](x) is the sesquilinear form in f and g associated with M. These results generalise the well-known theorem that M is in the limit-point case at ∞ if and only if [fg](∞) = 0 for every f,g∈ the maximal domain Δ associated with M.